New perspectives on self-linking

By Ryan Budney, James F. Conant, Kevin P. Scannell, Dev Sinha. 2003

70 citations recorded by the source corpus.

Abstract We initiate the study of classical knots through the homotopy class of the nth evaluation map of the knot, which is the induced map on the compactified n-point configuration space. Sending a knot to its nth evaluation map realizes the space of knots as a subspace of what we call the nth mapping space model for knots. We compute the homotopy types of the first three mapping space models, showing that the third model gives rise to an integer-valued invariant. We realize this invariant in two ways, in terms of collinearities of three or four points on the knot, and give some explicit computations. We show this invariant coincides with the second coefficient of the Conway polynomial, thus giving a new geometric definition of the simplest finite-type invariant. Finally, using this geometric definition, we give some new applications of this invariant relating to quadrisecants in the knot and to complexity of polygonal and polynomial realizations of a knot.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/af748f7eb5ab6c99dd2311b0dbd2685663d527b2

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Geometry and Topology

1. Dev Sinha
2. Ryan Budney
3. Victor Turchin
4. Victor Tourtchine
5. Ismar Volic
6. B. Munson
7. G. Arone

In Mathematical Physics

1. Dev Sinha
2. Ryan Budney
3. Victor Turchin
4. Ismar Volic
5. Victor Tourtchine
6. P. Lambrechts
7. B. Munson
8. G. Arone

Citations in the corpus

• Operads and knot spaces (2004, methodology): “For k ≤ 3, we may deduce this from the main results of [9].”
• Little cubes and long knots (2003, background): “Key words: spaces of knots; little cubes; operad; embedding; diffeomorphism”
• The topology of spaces of knots: cosimplicial models (2002, background): “The first is the mapping space model, which provides the proper setting for an evaluation map suitable for geometric study [10].”
• A family of embedding spaces (2006, background): “ [78] to verify that r generates H 2n 6(K n;1;Z). The following approach is perhaps more direct. It is inspired by the author’s quadrisecant description of the type-2 Vassiliev invariant for knots R3 [11]. The idea is to construct an integral co-homology class 2 2H2n 6(K n;1;Z) such that if x 2H 2n”